Generalized analytical model of Transient linear flow in heterogeneous fractured liquid-rich tight reservoirs with non-static properties

The industry is increasingly reliant on rate-transient analysis (RTA) to extract valuable information about the reservoir and hydraulic fractures. However, the application of current, commercially-available RTA models can lead to incorrect estimates of reservoir/fracture properties, potentially causing costly mistakes to be made in capital planning and reserve estimation. The root cause of these errors is that currently-available analytical solutions used in RTA models largely ignore reservoir heterogeneities, and assume static reservoir properties.In this work, a new transient linear flow is rigorously modeled in unconventional reservoirs with (1) pressure-dependent rock and fluid properties and (2) both continuous and discontinuous (heterogeneous) porosity and permeability. To achieve this, new transformations of pseudo-pressure, pseudo-time and pseudo-distance are first introduced to reduce the temporal and spatial non-linear diffusivity equation to that with approximately constant coefficients. Both a Laplace-domain solution and approximate analytical solution to the diffusivity equation are verified against a series of fine-grid numerical simulations for the assumption of fractal-based reservoir heterogeneity (over a wide range of stress-dependent rock and fluid properties). The results indicate that reservoir heterogeneity can result in nonlinear square-root-of-time plots. Further, rock and fluid pressure-dependencies act to decrease the slope of the square-root-of-time plot and affect reservoir/fracture property evaluations.Three liquid-rich shale (LRS) field examples in North America are analyzed to demonstrate the practical applicability of the new RTA models. Additional value of new RTA models over the sophisticated numerical simulation is to provide us an improved backforward-analysis workflow that can be used to quantify both effective fracture half-length and non-uniform permeability distribution around the fractures.The major contribution of this work is the introduction of a new analytical model for evaluating the transient linear flow period for the cases of arbitrary reservoir heterogeneity and non-static reservoir properties. This new approach is particularly useful for evaluating the effectiveness of hydraulic fracturing operations by extracting the spatial variability of reservoir quality within the stimulated reservoir volume (SRV).     

An analytical solution for conduction-dominated unidirectional solidification of binary mixtures

In this paper, a fully analytical solution technique is established for the solution of unidirectional, conduction-dominated, alloy solidification problems. By devising appropriate averaging techniques for temperature and phase-fraction gradients, governing equations inside the mushy region are made inherently homogeneous. The above formulation enables one to obtain complete analytical solutions for solid, liquid and mushy regions, without resorting to any numerical iterative procedure. Due considerations are given to account for variable properties and different microscopic models of alloy solidification (namely, equilibrium and non-equilibrium models) in the two-phase domain. The results are tested for the problem of solidification of a NH₄Cl–H₂O solution, and compared with those from existing analytical models as well as with the corresponding results from a fully numerical simulation. The effects of different microscopic models on solidification behaviour are illustrated, and transients in temperature and heat flux distribution are also analysed. A good agreement between the present solutions and results from computational simulation is observed.     

Analytical solution of a spatially variable coefficient advection–diffusion equation in up to three dimensions

Analytical solutions are provided for the two- and three-dimensional advection–diffusion equation with spatially variable velocity and diffusion coefficients. We assume that the velocity component is proportional to the distance and that the diffusion coefficient is proportional to the square of the corresponding velocity component. There is a simple transformation which reduces the spatially variable equation to a constant coefficient problem for which there are available a large number of known analytical solutions for general initial and boundary conditions. These solutions are also solutions to the spatially variable advection–diffusion equation. The special form of the spatial coefficients has practical relevance and for divergent free flow represent corner or straining flow. Unlike many other analytical solutions, we use the transformation to obtain solutions of the spatially variable coefficient advection–diffusion equation in two and three dimensions. The analytical solutions, which are simple to evaluate, can be used to validate numerical models for solving the advection–diffusion equation with spatially variable coefficients. For numerical schemes which cannot handle flow stagnation points, we provide analytical solution to the spatially variable coefficient advection–diffusion equation for two-dimensional corner flow which contains an impermeable flow boundary. The impermeable flow boundary coincides with a streamline along which the fluid velocity is finite but the concentration vanishes. This example is useful for validating numerical schemes designed to predict transport around a curved boundary.     

Piecewise finite series solutions of seasonal diseases models using multistage Adomian method

The aim of this paper is to apply the multistage Adomian Decomposition Method MADM to solve systems of nonautonomous nonlinear differential equations that describe several epidemic models with periodic behavior. Here the concept of the MADM is introduced and then it is employed to obtain a piecewise finite series solution. The MADM is used here as a hybrid analytical–numerical technique for approximating the solutions of the epidemic models. In order to show the efficiency of the method, the obtained numerical results are compared with the fourth-order Runge–Kutta method solutions. Numerical comparisons show that the MADM is accurate, easy to apply and the calculated solutions preserve the periodic behavior of the continuous models. Moreover, the method has the advantage of giving a functional form of the solution for any time interval. Furthermore, it is shown that if the truncation order and the time step size are not properly chosen large computational work is required and inaccurate solutions may be obtained.     

Analytical solution of the droplet breakup equation by the Adomian decomposition method

The droplet population balance equations (PBEs), which despite its importance rarely has an analytical solution. However, few cases with assumed functional forms of breakup rate, daughter droplet distribution exist, where most of these solutions are for the batch stirred vessel. A new framework for solving (PBEs) for batch and continuous systems is proposed in this work, which uses the Adomian decomposition method. This technique overcomes the crucial difficulties of numerical discretization and stability that often characterize previous solutions in this area. The technique used in this work has been tested for the droplet breakup equation. The solutions are presented for several cases, for which analytical solutions are available, for batch and continuous systems for droplet breakup in stirred vessels. In all cases, the predicted droplet size distributions converge exactly in a continuous form to that of the analytical solution.     

Approximate analytic solutions of stagnation point flow in a porous medium

An efficient and new implicit perturbation technique is used to obtain approximate analytical series solution of Brinkmann equation governing the two-dimensional stagnation point flow in a porous medium. Analytical approximate solution of the classical two-dimensional stagnation point flow is obtained as a limiting case. Also, it is shown that the obtained higher order series solutions agree well with the computed numerical solutions.     

Hybrid,numerical solutions,for three-dimensional,compressible Navier-Stokes layer

The author proposes new hybrid solutions for the three-dimensional, compressible Navier-Stokes layer (NSL) over a flying configuration (FC), which use the analytical potential solutions, of the same FC, two times, namely: to reinforce the numerical solutions by multiplying them with these analytical solutions and as outer flow (instead of the parallel flow, used by Prandtl in his boundary layer theory). These hybrid solutions fulfill the last behavior, have correct jumps along the singular lines (like subsonic leading edges, junction lines wing-fuselage, etc.), are split, accurate and rapid convergent. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Analytical solution of three-dimensional Navier–Stokes equations for the flow near an infinite rotating disk

In this letter, we will consider variational iteration method (VIM) and Padé approximant, for finding analytical solutions of three-dimensional viscous flow near an infinite rotating disk. The solutions is compared with the numerical (fourth-order Runge–Kutta) solution. The results illustrate that VIM–Padé is an appropriate method in solving the systems of nonlinear equations. It is predicted that VIM–Padé can have wide application in engineering problems (especially for boundary-layer and natural convection problems).     

Oscillation waves in Riemann problems inside elliptic regions for conservation laws of mixed type

Riemann problems with initial data inside elliptic regions are quite different from those for hyperbolic systems. First, we have found that approximate solutions may present persistent oscillations, giving rise to a new type of (measure-valued) waves besides the usual (distributional) ones, shocks and rarefaction waves. Second, any local disturbance of a constant state inside the elliptic region will result in a non-trivial (distributional or, more generally, measure-valued) solution, which is independent of any particular choice of disturbance. For our numerical experiments, we establish two analytical results for testing convergence of finite difference schemes, and for determining expectation values of state functions with respect to the measure-valued solutions when oscillation waves occur. Numerical examples are presented to illustrate those interesting aspects, including the appearance of oscillation waves together with the analysis of the corresponding Young measures.     

Analytical solutions of the boundary layer flow of power-law fluid over a power-law stretching surface

This article discusses analytical solutions for a nonlinear problem arising in the boundary layer flow of power-law fluid over a power-law stretching surface. Using perturbation method analytical solution is presented for linear stretching surface. This solution covers large range of shear thinning and shear thickening fluids and matches excellently with the numerical solution. Furthermore, some new exact solutions are found for particular combination of m (power-law stretching index) and n (power-law fluid index). This leads to generalize the case of linear stretching to nonlinear stretching surface. The effects of fluid index n on the boundary layer thickness and the skin friction for nonlinear stretching surface is analyzed and discussed. It is observed that the boundary layer thickness and the skin friction coefficient increase as non-linear parameter n decreases. This study gives a new dimension to obtain analytical solutions asymptotically for highly nonlinear problems which to the best of our knowledge has not been examined so far.     

Multiple solutions of a boundary-value problem in enzyme kinetics

In some immobilized enzyme systems the steady state of substrate concentration may suddenly change from a low profile to a high profile or vice versa when the physical parameters of the systems pass through certain critical values. This phenomenon is due to the transition from a unique solution to multiple solutions (or vice versa) of the enzyme reaction equation. This problem is studied by considering two physical parameters which represent the internal reaction mechanism and the external influence on the boundary of the reaction-diffusion medium. Both analytical and numerical results for the problem are presented. The analytical results include some sufficient conditions for the existence of multiple steady-state solutions as well as a unique solution. Various numerical results of the problem including time-dependent solutions and their convergence to steady-state solutions are given.     

Use of 3D-transient analytical solution based on Green’s function to reduce computational time in inverse heat conduction problems

Inverse problems can be found in many areas of science and engineering and can be applied in different ways. Two examples can be cited: thermal properties estimation or heat flux function estimation in some engineering thermal process. Different techniques for the solution of inverse heat conduction problem (IHCP) can be found in literature. However, any inverse or optimization technique has a basic and common characteristic: the need to solve the direct problem solution several times. This characteristic is the cause of the great computational time consumed. In heat conduction problem, the time consumed is, usually, due to the use of numerical solutions of multidimensional models with refined mesh. In this case, if analytical solutions are available the computational time can be reduced drastically. This study presents the development and application of a 3D-transient analytical solution based on Green’s function. The inverse problem is due to the thermal properties estimation of conductors. The method is based on experimental determination of thermal conductivity and diffusivity using partially heated surface method without heat flux transducer. Originally developed to use numerical solution, this technique can, using analytical solution, estimate thermal properties faster and with better accuracy.     

A finite element method for steady-state conduction-advection phase change problems

A new finite element: technique is developed to solve steady-state conduction-advection problems with a phase change. The energy balance equation at the solid/liquid interface is employed to calculate the velocity of the solid/liquid interface in the Lagrangian frame. The position of the solid/liquid interface in the Eulerian frame is determined based on the composition of the velocity of the solid/liquid interface in the Lagrangian frame and the steady-state velocity of a rigid body. The interface position and the finite element mesh are continuously updated during an incremental process. No artificial diffusion is needed with this new finite element approach. An analytical solution for solidification of a pure material with a radiative boundary condition is also developed in this paper. Numerical experimentation is conducted and the corresponding results are compared with analytical solutions. The numerical results agree well with analytical solutions.     

Fourier–Bessel theory on flow acoustics in inviscid shear pipeline fluid flow

Flow acoustics in pipeline is of considerable interest in both industrial application and scientific research. While well-known analytical solutions exist for stationary and uniform mean flow, only numerical solutions exist for shear mean flow. Based on potential theory, a general mathematical formulation of flow acoustics in inviscid fluid with shear mean flow is deduced, resulting in a set of two second-order differential equations. According to Fourier–Bessel theory which is orthogonal and complete in Lebesgue Space, a solution is proposed to transform the differential equations to linear homogeneous algebraic equations. Consequently, the axial wave number is numerically calculated due to the existence condition of non-trivial solution to homogeneous linear algebraic equations, leading to the vanishment of the corresponding determinant. Based on the proposed method, wave propagation in laminar and turbulent flow is numerically analyzed.     

A posteriori error analysis of round-off errors in the numerical solution of ordinary differential equations

We prove sharp, computable error estimates for the propagation of errors in the numerical solution of ordinary differential equations. The new estimates extend previous estimates of the influence of data errors and discretization errors with a new term accounting for the propagation of numerical round-off errors, showing that the accumulated round-off error is inversely proportional to the square root of the step size. As a consequence, the numeric precision eventually sets the limit for the pointwise computability of accurate solutions of any ODE. The theoretical results are supported by numerically computed solutions and error estimates for the Lorenz system and the van der Pol oscillator.     

Fully–coupled model for electrokinetic flow and transport in microchannels

We investigate the electrokinetic flow and mass transport in microchannels. Therefore, mathematical models of the electrical, fluid-mechanical and chemical processes are established. Within the electrical double layer (EDL), approximative analytical solutions can be found and matched asymptotically to the numerical (FEM) solution of the channel core. The results of the simulations show a strong coupling between the processes. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

多机器人协调吊运系统逆运动学分析及优化

针对紧耦合多机器人协调吊运系统的逆运动学问题进行了分析,首先利用几何关系和力旋量平衡方程建立了系统的运动学模型和动力学模型;然后对系统的逆运动学进行分析,将其分为变柔索长度和固定柔索长度两种情况分别进行分析;随后对运动学逆解在某一时刻存在无穷多解、多组解和无解的情况分别给出了解决方法,对存在多组解的情况,提出一优化目标求解最优解;最后结合软件UG/ADAMS/MATLAB建立了系统的实验平台,通过实例仿真计算验证了方法的有效性,为后续进一步研究系统运动稳定性、优化拉力分布和控制算法奠定了基础.     

Competing through altering the environment: A cross-diffusion population model coupled to transport–Darcy flow equations

We consider an evolution model describing the spatial population distribution of two salt tolerant plant species, such as mangroves, which are affected by inter- and intra-specific competition (Lotka–Volterra), population pressure (cross-diffusion) and environmental heterogeneity (environmental potential). The environmental potential and the Lotka–Volterra terms are assumed to depend on the salt concentration in the root region, which may change as a result of mangroves’ ability to uptake fresh water and leave the salt of the solution behind, in the saturated porous medium. Consequently, partial differential equations modelling the population dynamics on the surface are coupled with Darcy–transport equations modelling the salt and pressure-velocity distribution in the subsurface. We prove the existence of weak solutions of the coupled problem and provide a numerical discretization based on a stabilized mixed finite element method, which we use to numerically demonstrate the behaviour of the system.     

New analytical and CWENO numerical solutions for axisymmetric horizontal flows with heat sources

Some new nonlinear analytical solutions are found for axisymmetric horizontal flows dominated by strong heat sources. These flows are common in multiscale atmospheric and oceanic flows such as hurricane embryos and ocean gyres. The analytical solutions are illustrated with several examples. The proposed exact solutions provide analytical support for previous numerical observations and can be also used as benchmark problems for validating numerical models. A central weighted essentially non-oscillatory (CWENO) reconstruction is also employed for numerical simulation of the corresponding integro-differential equations. Due to the use of the same polynomial reconstruction for all derivatives and integral terms, the balance between those terms is well preserved, and the method can precisely reproduce the exact solutions, which are hard to capture by traditional upwind schemes. The developed analytical solutions were employed to evaluate the performance of the numerical method, which showed an excellent performance of the numerical model in terms of numerical diffusion and oscillation.     

On a new numerical algorithm for the solution of triple deck problems

A new algorithm for the solution of laminar viscous-inviscid interactions allowing for the formation of pronounced separation zones is presented. It is then applied to two different types of flows: near critical two layer fluid flow and subsonic flow past an expansion ramp. In the first case emphasis is placed on the formation of so-called non-classical hydraulic jump having the distinguishing property that waves pass through rather than merge with the jump. The second problem exhibits the interesting feature that no solution exists if the ramp angle exceeds a critical value while two solutions exists if the ramp angle is subcritical. The new numerical scheme is used to study the change in the flow behaviour as the critical ramp angle is approached. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)